Twisted cubic and point-line incidence matrix in $\mathrm{PG}(3,q)$

TL;DR

This paper investigates the structure of the point-line incidence matrix in projective space PG(3,q), focusing on orbits under the twisted cubic's stabilizer group, and derives detailed incidence counts for various line and point configurations.

Contribution

It provides a detailed analysis of the incidence matrix structure related to the twisted cubic in PG(3,q), including the splitting of submatrices and enumeration of incidences for different line orbits.

Findings

Explicit incidence counts for points and lines in submatrices.

Decomposition of incidence matrices into smaller structured blocks.

Characterization of line and point incidences for specific line orbits.

Abstract

We consider the structure of the point-line incidence matrix of the projective space $\mathrm{PG}(3,q)$ connected with orbits of points and lines under the stabilizer group of the twisted cubic. Structures of submatrices with incidences between a union of line orbits and an orbit of points are investigated. For the unions consisting of two or three line orbits, the original submatrices are split into new ones, in which the incidences are also considered. For each submatrix (apart from the ones corresponding to a special type of lines), the numbers of lines through every point and of points lying on every line are obtained. This corresponds to the numbers of ones in columns and rows of the submatrices.